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# 33. Quaternions

Quaternions are hypercomplex numbers used to represent spatial rotations in three dimensions. This set of routines provides a useful basis for working with quaternions in Octave. A tutorial is in the Octave source, scripts/quaternion/quaternion.ps.

These functions were written by A. S. Hodel, Associate Professor, Auburn University.

Function File: [a, b, c, d] = quaternion (w)
Function File: [vv, theta] = quaternion (w)
Function File: w = quaternion (a, b, c, d)
Function File: w = quaternion (vv, theta)
  Construct or extract a quaternion

      w = a*i + b*j + c*k + d

  from given data.

Function File: qconj (q)
  Conjugate of a quaternion.

      q = [w, x, y, z] = w*i + x*j + y*k + z
      qconj (q) = -w*i -x*j -y*k + z

Function File: qderiv (omega)
  Derivative of a quaternion.

  Let Q be a quaternion to transform a vector from a fixed frame to a rotating frame. If the rotating frame is rotating about the [x, y, z] axes at angular rates [wx, wy, wz], then the derivative of Q is given by

      Q' = qderivmat (omega) * Q

  If the passive convention is used (rotate the frame, not the vector), then

      Q' = -qderivmat (omega) * Q

Function File: qderivmat (omega)
  Derivative of a quaternion.

  Let Q be a quaternion to transform a vector from a fixed frame to a rotating frame. If the rotating frame is rotating about the [x, y, z] axes at angular rates [wx, wy, wz], then the derivative of Q is given by

      Q' = qderivmat (omega) * Q

  If the passive convention is used (rotate the frame, not the vector), then

      Q' = -qderivmat (omega) * Q.

Function File: qinv (q)
  Return the inverse of a quaternion.

      q = [w, x, y, z] = w*i + x*j + y*k + z
      qmult (q, qinv (q)) = 1 = [0 0 0 1]

Function File: qmult (a, b)
  Multiply two quaternions.

      [w, x, y, z] = w*i + x*j + y*k + z

  identities:

      i^2 = j^2 = k^2 = -1
      ij = k                 jk = i
      ki = j                 kj = -i
      ji = -k                ik = -j

Function File: qtrans (v, q)
  Transform the unit quaternion v by the unit quaternion q. Returns `v = q*v/q`.

Function File: qtransv (v, q)
  Transform the 3-D vector v by the unit quaternion q. Return a column vector.

      vi = (2*real(q)^2 - 1)*vb + 2*imag(q)*(imag(q)'*vb)
         + 2*real(q)*cross(imag(q),vb)

  Where imag(q) is a column vector of length 3.

Function File: qtransvmat (qib)
  Construct a 3x3 transformation matrix from quaternion qib that is equivalent to rotation of th radians about axis vv, where `[vv, th] = quaternion (qib)`.

Function File: qcoordinate_plot (qf, qb, qv)
  Plot in the current figure a set of coordinate axes as viewed from the orientation specified by quaternion qv. Inertial axes are also plotted:

  qf
    Quaternion from reference (x,y,z) to inertial.
  qb
    Quaternion from reference to body.
  qv
    Quaternion from reference to view angle.

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